Monthly Archives: May 2014

It sure is a number line, and it works perfectly well with the whole or natural numbers.

The question is “How did the number line become straight, with equally spaced numbers, when the ideas of length and measurement have not yet been developed?”. This is the math version of the “what came first, the chicken or the egg?” question.

CCSS grade 6Apply and extend previous understandings of multiplication anddivision to divide fractions by fractions.How many 3/4-cup servings are in 2/3 of a cup of yogurt?

Well, I would say “None”

CCSS grade 7Analyze proportional relationships and use them to solve real-worldand mathematical problems.1. Compute unit rates associated with ratios of fractions, including ratiosof lengths, areas and other quantities measured in like or differentunits. For example, if a person walks 1/2 mile in each 1/4 hour, computethe unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.

And here we are asked to crack a nut with a sledgehammer.Common sense to the rescue!Half a mile in a quarter of an hour is one mile in half an hour and so 2 miles in one hour, or 2 miles per hour.

1/2—-1/4is a complex fraction …. really? Only if fractions are not numbers!!!!!!!

and it contains what before 1984 we would have called a 1984 scenario:

“In this mode of instruction, these devices become the mediator of almost every academic interaction between students and their teacher, and even one another. Students are assigned work on the device, they perform their work on the device, they share work through the device, and they receive feedback via the device. What is more, the means by which learning is measured—the standardized test—will also be via this device.”

So, it is a classroom in the future.

“Now, children, we are going to measure the classroom. How many Ipads long is the room? And how many wide?”

These are signs or symbols for numbers. they are NOT numbers, and they are not unique –

ten is 1010 in the binary system

A in the hexadecimal system

X in the Roman system

and 10 only in the decimal system.

Of course, sooner or later we treat the symbolic form as “the number” , and it works, but we should never forget that symbols are not numbers.

The basic numbers, or natural numbers (as mathematicians call them).

These are one, two, three etc, and including zero as it is very useful, and are for COUNTING and COMPARING quantities of things.

“How many peas in this pod?”

“How many sweets in this bowl?”

Also for resolving issues like “You have got more sweets than I have!”

Operations with the natural numbers.

Combining quantities leads to “addition”.

Comparing quantities leads to “subtraction”.

Combining like sized groups of things leads to “multiplication”.

Sharing leads to “division”.

Each of these processes needs to be thoroughly explored and understood in words before inflicting symbolic notation on the students.

In passing I have to comment on the term “word problem”, which I have encountered very frequently. To me, teaching engineers, computer scientists and others a problem is a problem. A much more satisfactory classification is “real problems” and “symbolic problems”, and mathematics is about both of these. The term “word problem” is at best misleading and at worst downright stupid.

It should be noticed that with addition and subtraction of natural numbers that the objects concerned must be of the same type, but this is NOT the case with multiplication and division. Consequently the order is irrelevant with addition, but this is definitely not obvious with multiplication.

While working on my software for 3D spline curves I needed to find a point between two others which was not halfway.

This turned into finding the ratio which an angle bisector of a triangle splits the opposite side. I worked it out with coordinate geometry and vectors, messy, messy, and then found out that this was a regular theorem in geometry. Here is the geometrical proof which I came up with. It sure ties a few things together:

I often wondered about the continued existence of the division sign. Not only is it redundant, it is confusing. A search of the net gave me this:

Part of the IT standards and organizations glossary: The division sign resembles a dash or double dash with a dot above and a dot below (÷). It is equivalent to the words “divided by.” This symbol is found mainly in arithmetic texts at the elementary-school level. It is rarely used by professional or academic mathematicians, scientists, or engineers.

Well, even though the outside world (the real world) uses / (as in spreadsheets and computer keyboards)or “dividend” over “divisor”, as in formulae and algebra generally, my guess is that this looks too much like a fraction !!!!! (which of course is not a number, or at best a very weird number).